
For some families of reward distributions, it is possible to compute \(\pi _{t,1}^{\text{TS}}\)  either analytically or by quadrature. However, due to a potential computational burden,
the typical way of implementing TS [1]} avoids the direct computation of such probabilities and proceeds as follows. Assuming a prior distribution for the unknown parameters \((\mu _0, \mu _1)\) , at each step \(t \ge 1\)  a sample is drawn from the updated posterior distributions, say \((\tilde{\mu }_{t,0}, \tilde{\mu }_{t,1})\)  and the arm \(\tilde{a}_{t,i}\)  with the highest estimated posterior mean reward \(\mathbb {\tilde{E}}(Y_{t,i}|A_{t,i} = k) = \tilde{\mu }_{t,k}\)  is allocated. That is:
\(\tilde{a}_{t,i} \doteq \text{argmax}_{k = 0, 1}\mathbb {\tilde{E}}(Y_{t,i}|A_{t,i} = k) = \text{argmax}_{k = 0,1}\tilde{\mu }_{t,k}.\) 
